Tuesday, 8 December 2009

A raw first draft of some thought on scepticism, a belief in an absolute objective truth and relativism. The use of logical notation is only partly so I can fuel fantasies about being a theoretical physicist, it's also because a lack of skill in writing prevents me from rendering somethings in halfway decent English.Whilst the absolutist and relativist agree that “the world is everything that is the case”i they disagree on what they take to be “the case”. They will, naturally, often agree on specifics. Both the relativist and the absolutist are likely to hold the same opinion on the correctness of an uncontroversial statement such as “there is a blue Everton shirt in front of Eve” (call this statement “s”). Their disagreement is more abstract: over what it means for something to be “the case”, what causes it to be so, what makes s true.

Absolutism

The absolutist holds that what is the case simply is the case, utterly independent of the observer. There either is a blue Everton shirt in front of Eve or there is not. s is true just in those cases where there is a blue Everton shirt in front of Eve and false just in those cases where there is not a blue Everton shirt in front of Eve.

s entails a number of other putative facts, that there is an object that absorbs red light in front of Eve, that the object is made of cloth, that the object has a definable badge with Everton's motto “nil satis nisi optimum” and so on. Other statements, not identical to s, may have the same entailments. These same entailments may be expressed in a different language, may be expressed by speech or text, will expressed differently even if spoken in the same language in a strong Liverpool accent as opposed to in received pronunciation. “Il y a un maillot Everton en face d'Eve” and “there is a blue Everton shirt in front Eve” both entail that there is an object that absorbs red light, that the object is made of cloth and so on. Although they are different statements it can be said that they express the same proposition, from here on referred to as p.

Any two series of statements that purport to be a complete description of what is the case will either express the same or differing propositions. If the series express the same proposition then all statements that the series entail will express the same propositions. A complete description of reality in French will, if it contains the statement “il y a un maillot blue Everton en face d' Eve” entail that there is an object that absorbs red light. A complete description of reality in English will, if it contains the statement “there is a blue Everton shirt in front of Eve”, also entail that there is an object that absorbs red light. Now there either is, or is not, at a specific time and place an object that absorbs red light. If our series of statements in French is true it will correspond to that state of affairs: affirming it if it is the case, denying it if it is not the case. If both our French and our English series of statements are to be true then they must both entail the same proposition about the presence of red light absorbing objects. If they are true, and complete, descriptions of what is the case then for each aspect of what is the case they must agree with each other as well as correspond to what is the case. If they are both true they must express the same set of propositions, what is the case corresponds to just the one set of propositions: there is only one totally of what is the case. There is just the one world.

Relativism

Most relativists think this is all hopelessly vague, often circular and ultimately unproductive. The proposition seems particularly vague. A “proposition” seems to be, more or less, what a statement is if you remove all the language: and that sounds like silence. It we wish to give the concept some content we have to talk in term of what is the case rather than our method of talking about it. This makes the concept of truth sound circular, propositions are the “what is the case” about a statement that statement being true if “what is the case” corresponds to what is the case. Whether circular or not it is unproductive, to operate the concept in our lives we cannot leave the world and truth as ineffable mysteries. We need true statements and knowledge of hard facts and that necessitates that whatever is the case is determined.

This determination necessitates a framework; an “ideology”, “paradigm”, “discourse”, “world view”, “set of presuppositions” and the like. Frameworks vary, producing differing determinations of what is the case and there is no framework independent method of determining which framework is “correct”. A determination of what is the case is, thus, relative to the framework that determined that it is the case. Whilst a statement can be correct just in so far as it corresponds to what is the case it makes little sense to talk of a statement being correct per se. Eve's statement s is right relative to Eve's framework just in case Eve's framework determines that there is a blue Everton shirt in front of her and wrong just in case Eve's framework does not determine that there is a blue Everton shirt in front of her. (For ease of distinction in this paper the terms “true” and “false” will be used to denote objective truth and falsehood, the terms “right” or “wrong” will be reserved for correspondence with relatively determined facts.)

Assume a situation, call it “situation1” where we would assent to s. We would assent to s because s emanates from a framework that uses the concepts of “Everton”, “shirt” and “blue”. Football clubs are a cultural construct as is the concept of “shirt”. The idea of a football club specific shirt is another cultural gift, allowing some shirts to win first prize in the lottery of shirt-dom and be Everton shirts. The blue arises as result of the human eye and brain which compartmentalises the smooth variations in wavelengths of light into distinct colours.

And none of this need be so. Were we in the early 19th century, if we had a 19th century framework, then “football” would denote a very different activity than it does today. We have to wait until 1878 before there is an Everton Football Club for the shirt to pertain to, whilst still earlier frameworks would fail to recognise a shirt. Other cultures extant today, despite the march of globalisation, would not recognise a shirt, or football, or Everton. The French “il y a un maillot blue en face d'Eve” is not entirely commensurate with the English. The French use the word “maillot” for an item of sportswear, whereas “shirt” covers both sportswear and everyday clothing. You can wear a shirt to work but not a maillot. The blueness of the shirt arises, if we accept what we are told, because it absorbs most of the visible spectrum of light reflecting only the portion we call “blue”. Were the shirt to reflect both the blue and green portions it would not be both blue and green but yellow. Now humans only see a small range of wavelengths of light, insects see more: they see ultra-violet. Where an object absorbs all light except for ultra-violet an insect will see an ultra-violet object and a human a black object. If the putative Everton shirt reflected back both blue and ultra-violet the insect would see a different colour that, because we have never seen it, we have no name for. Remaining within the bounds of human frameworks we may use Goodman's terms of grue and bleen to describe the shirt. A shirt is grue if, when examined before time t, it is green and blue if unexamined. A bleen shirt is one that is blue if examined before time t and green if unexamined.ii Or we may be speakers of one of those human languages that do no have a term that corresponds to blue. s can only be right within a very particular framework, one which there is no objective justification for privileging.

Frameworks vary, producing differing determinations of what is the case. A judgement between frameworks itself dependent on a framework. As a result although framework-f may be judged the absolute business within framework-f, outside of any framework it cannot be judged more (or less) valid than any other. The necessity of frameworks renders all frameworks equally valid.

This much, the doctrine of equal validity, is a common factor in relativism. The next step brings a divergence. Some insist that a framework is not just necessary but sufficient for establishing what is the case, the doctrine that frameworks are “equally valid and valid”. The view manifests itself in popular discourse with such ideas that there are “different ways of knowing the world”: the differing ways result in knowledge of what is the case. As they are differing ways they result in differing worlds.

An alternative take is to deny that a framework is ever sufficient. Any determination of what is the case must transcend a framework which, as the framework is necessary, is an impossibility. We are left with various “discourses” that we tale part in all with little, or no, relation to anything that might be said to be “the case”.

To bring the strands of relativism together and for ease of distinction from absolutism, the terms “true” and “false” will be used to denote objective truth and falsehood, the terms “right” or “wrong” will be reserved for correspondence with relatively determined facts.iii

Partial Refutation of Relativism

Whilst it is entirely correct that nothing that is the case can be determined it is incorrect that nothing can be determined about what is the case.

Take a situation (call this “situation2”) where the putative blue Everton shirt in front of Eve is replaced with something it would be right to call a red pair of Liverpool shorts. s has now become wrong. It becomes wrong independent of any framework: our conclusion changes, but our framework has not. We have not changed culture, developed new cognitive apparatus or undergone a sudden conversion experience. The world has changed and with it our conclusion about the world.

Neither does a person with an alternative framework, one which fails to support s when we maintain it, fail to reject s when we reject it. The framework is a necessary condition of s being right. If s is right then our framework, or the relevant part of it, holds for the observer. If our framework fails to hold for Trevor then s is not right for Trevor whether or not somebody is waving a pair of Liverpool shorts in his face. Nor does our framework need to support s being wrong. If we is presented with black Manchester United shorts, green Norwich City socks or a pint of Real Ale then s is wrong. It is not necessary to establish what it is that is in front of Eve, just that it is not a blue Everton shirt. It is not what is the case that makes s wrong but what is not the case.

Resisting the temptation to describe what is the case, a description that depends on a framework and is subject to relativist objections, we can designate what fails to be the case as a. We can accept that we have no idea what a is and accept that we have no idea what is the case in situation2. We must also accept both that a is a consequence of s ( s→a)iv and that a does not obtain, independent of any framework (¬a). Thus s is not just wrong, it is false. In situation2 a does not obtain, but it is entailed by s. If s were true then a would obtain, a does not obtain, thus s is false.

We can now make the confusing sentence that began this section clear. “Nothing that is the case can be determined”: we cannot say what is the case. “Incorrect that nothing can be determined about what is the case”: we can often say some of what is not the case.

If there where no objective states of affairs to which propositions may correspond no statement could be refuted solely on manipulation of what is the case. Statements can be refuted solely on manipulation of what is the case, thus there is an objective state of affairs to which propositions may correspond.

Sceptical Truth-schemaResisting the temptation to describe what is the case, a description that depends on a framework and is subject to relativist objections, we can designate what fails to be the case as a. We can accept that we have no idea what a is and accept that we have no idea what is the case in situation2. We must also accept both that a is a consequence of s ( s→a) and that a does not obtain, independent of any framework (¬a). Thus s is not just wrong, it is false. In situation2 a does not obtain, but it is entailed by s. If s were true then a would obtain, a does not obtain, thus s is false.

Of course a is just one of a whole host of consequences of s, any one of which is sufficient to render s false. If there exists any x such that s entails x and x is false then s is false:

1.∃x [(s→x) & ¬x] → ¬s(if there exists an x which is entailed by s and that x is false then s is false)

Now if there is an x entailed by s that is capable of being true or false then it is either true or false and thus everything entailed by s that is capable of being true or false is either true or false. If everything entailed by s were true then s would be true. To be false s must make a false claim. The falsity of s entails that there will be something false entailed by s:

2.¬s → ∃x [(s→x) & ¬x](if s is false then there will exist an x entailed by s that is false)

This is equivalent to:

3.∃x [(s→x)] & ¬∃x [(s→x) & ¬x] → sv(if there exists an x, such that x is entailed by s and there exists no x such that s is is entailed by x and is false then s is true)

For good measure “3.” is equivalent to:

4.s → ¬∃x [(s→x) & ¬x](if s is true then there does not exist an x such that x is entailed by s and x is false)

Which combines with “3.” to give:

5.s ↔ ∃x [(s→x)] & ¬∃x [(s→x) & ¬x](s is true if and only if there exists an x, such that x is entailed by s and there exists no x such that s is is entailed by x and is false)

The schema holds that entailment combined with an absence of false entailments is both necessary and sufficient for the truth of a statement. The combination of necessity and sufficiency negates both the necessity of a framework and the sufficiency of a framework.

We are not entitled to conclude from the premise that a statement either entails nothing or entails something that is false that the statement is not correctly made from within a framework. There are plenty of examples of statements made perfectly in accordance with an epistemology, all the relevant data, in accordance with a preferred ideology that have predicted events that did not come to pass. A framework is not sufficient.

As for the necessity, the schema defines a logical relation between a statement and the world it purports to describe. A framework may be contingently necessary for a statement to entail, but it is not a logical necessity. Neither, as we have seen above, is a framework necessary for statement to fail to conflict with reality. Where there is contingency in failing to conflict it is a contingency of the world rather than the statement.be false.

It has to be said, however, that the contingent need for a framework almost universally obtains. The relativist arguments against the absolutist hold for all actual statements made about the world. We cannot say anything true about the world if we use a framework, we cannot say anything about the world without a framework. It follows that we cannot say anything true about the world.

Friday, 12 June 2009

Have I gone mad? Taking a favoured tool of those who accept induction to argue against them? I don’t believe I have gone mad, this is quite calmly considered. I am well aware, however, that I am writing using something (maths) I do not have a full grasp of. This may not be madness but it is fertile ground for internet crankness. Internet cranks start off making simple, basic, errors in areas they do not fully understand and, on the basis of these errors, leap to wild conclusions. Now I haven’t quite gone for other aspects of internet crankery, branding anyone who disagrees with me as part of a conspiracy or cackling madly. But I am worried that what follows contains a really basic error. If it does I would be very grateful to anyone who pointed it out to me.Induction and Abduction

Take a number of hypotheses that make a prediction, one way or another, about a piece of evidence “E”:

1. H1, H2, H3 and H4 which have been formulated and2. A number of hypotheses H5 to HN which haven’t been formulated yet, for brevity let’s call all these “HU”

Before anything, evidence, abduction or induction we allocate a range of probabilities over these hypotheses.

1. We may firmly believe H1: “H1 is true”.2. We may spread the probability over the four formulated hypotheses: “Either H1, H2, H3 or H4 are true” or3. we may decide that we should include the unfortmulated propostions: “either H1, H2, H3, H4 or HU is true”.

Whatever distribution of probabilities we decide on the sum of all these is not dependent on that distribution. If we reject a hypothesis we reduce the probability, we assign it to zero. To keep the sum of probabilities stable we must increase the probability assigned to the remaining hypotheses. Without a principle, inductive evidence or prejudice to guide us we are free to distribute this increase in probability how we wish over the non-rejected propositions. That is to say that no particular proposition is singled out as being more likely to be true. All that we are forced to is a conjunction of the hypotheses left. This is abduction and is deductive:

1. Either H1, H2, H3 or H42. It’s not H33. Thus either H1, H2 or H4

is perfectly valid.

Induction, however, confirms a proposition. It does pick out a specific hypothesis and make that hypothesis more likely. If we have evidence for, say, H2 then we must increase the probability we assign to H2. If we have evidence against H3 then we could redistirubute all that probability to H1 (say we started off believing "it's either H1 or H3" and falsified H3, we'd then believe "it's definitely H1"). If we have evidence that is solely supportive of H2 then we would have to increase the probability we assign to E2 alone.

Induction, evidence for, picks out a hypothesis, or group of hypotheses, and increases the likelihood we should attach to them. How we spread the necessary reduction in other probabilities is up to us. Abduction, evidence against, picks out a hypothesis or group of hypotheses and decreases the likelihood we should attach to them. How we spread the necessary increase in other probabilities is up to us.

What I hope to show is that any movement in the probability assigned to a hypothesis is either abductive or unrelated to the evidence. If I show this I will consider that I have shown that induction does not exist.

Bayes Theorem

To show this I am using the arguments of those who believe in induction against them. There current favourite tool is Bayes’ Theorem which states:

P(H¦E) = P(E¦H) P(H) / P(E)

Where:

P(H¦E) = The probability of a hypothesis ,“H”, given an item of evidence “E”

P(E¦H) = The probability of the evidence given the hypothesis

P(H) = The probability of the hypothesis before considering the item of evidence (the “prior probability”)

P(E) = the probability of the evidence arising (without direct reference to the hypothesis)

(For an explanation of Bayes’ theorem that even I can understand (after two or three readings) go to Eleizer Yudkowsky’s )

Competing Predicitions

Return to the example list of hypotheses above and lets say that H1 and H2 predict E whilst H3 and H4 predict not-E. Not having formulated HU we can’t tell whether those hypotheses predict or forbid E.

We then see E.

This, naturally, should make us downgrade our belief in H3 and H4. It follows that we should increase the level of probability assigned to “H1, H2 and HU” but says nothing about how we should now distribute that probability across these hypotheses. So far, so abductive. This is because we have just considered the negative effect on H3 and H4 and not the positive, inductive, support that E gives any one of H1, H2 or HU.

Equal Predictions

How does E inductively effect, say, H1? It’s no good showing that the probability of H1 after the evidence, P(H1¦E), is greater than the probability of H1 before the evidence, P(H1). We know that already by abduction. To show inductive support we need to show that the P(H1¦E) is greater than the probability of another unfalsified propostion, say H2. We need to show that P(H1¦E) > than P(H2¦E) or, from Bayes’ Theorem:

P(E¦H1) P(H1) / P(E) > P(E¦H2) P(H2) / P(E)

P(E) is on both sides, so we can cancel it out:

P(E¦H1) P(H1) > P(E¦H2) P(H2)

In this part of the argument I am assuming that both equally predict the evidence, thus P(E¦H1) = P(E¦H2) and they can be cancelled out:

P(H1) > P(H2)

So our comparison of posterior probabilities depends upon the prior probabilities. How could they be different? If it is absolutely nothing to do with prior evidence then on the evidence P(H1) = P(H2) and thus P(H1¦E) = P(H2¦E). So any evidential support of E for H1 over and above H2 depends on prior evidential support for H1 over H2. This can’t be abductive, because this will increase the probability of H1 and H2 without favouring either. So the effect of the prior evidence on the prior probabilities depend on the prior-prior probabilities going into that particular calculation. The regress will go back with differentials of prior probabilities depending on previous prior probabilities until we get back to before any evidence whatsoever. If, and only if, a differential here depends on evidence will any of the later differentials in prior probabilities depend on evidence. Of course a differential that depends on evidence arising before evidence is an absurdity. Thus, where hypotheses predict evidence with equal probability any support beyond that given by abduction, is not from evidence but merely the reinforcement of non-evidential belief.

Unequal Predictions

If on the evidence P(H1) = P(H2) then P(E¦H1) > P(E¦H2) will result in P(H1¦E) > P(H2¦E).

However any hypothesis that predicts evidence with probability of less than 1 can be readily converted to a hypothesis that gives a definite prediction by adding a hypothesis about the probability predictions. If H2 predicts E will occur one in two times then the following formulations of H2* will predict that E will occur all the time:

H2* : H2 and “whatever combines with H2 to produce E” or

H2* : H2 and “it just happens to be one of those times when H2 does produce E”

The same argument can be applied to H1, to create H1* (or “H1 and W1”) which makes a definite prediction of E. If we do that then, from the analysis of hypotheses that predict the evidence with equal certainty, we know that evidence cannot favour one over the other. So there cannot be any evidential support of H1* over H2*. Can we assess the evidential support of H1 and H2 without the additional factor?

No. Because, given E, “H2 and not-W2” is falsified. Thus “H2 and W”, or H* is the only “live” hypothesis. The same argument applies to “H1 and W1”. This gets us back to the equality (on the evidence) of prior probabilities and no variation (on the evidence) of posterior possibilities. Thus evidence fails to favour one hypothesis over any other non-falsified hypothesis. Induction does not exist.

Friday, 27 February 2009

I have been mulling over this particular issue for a long time and, as it popped up on Stephen Law's blog, I decided to get my thoughts down.

Alvin Plantinga's argument from evolution against naturalism is an ingenious and controversial argument. Plantings argues that evolution without the intervention of a supernatural being makes our beliefs very unlikely to have been reliably formed. One of those beliefs would be evolution without the intervention of a supernatural being. Evolutionary naturalism is therefore "self defeating". Do panic at the word "evolution", the argument is not some idiot creationist nonsense. It is very well formed, very well argued and examining it brings interesting insights into defeaters and what is means to have a rational belief. "Undercutting" defeaters are themselves self-defeating.

A brief summary of the argument The argument starts with pointing out that if there was no intervention from a supernatural being in the evolution of man then we are entirely reliant on the operations of evolution for the production of reliable belief-forming mechanisms, reliable cognitive processes. Plantinga then questions whether evolution gives two hoots about our cognitive processes. If we behave right then we will be evolutionarily successful, whether or not we believe right. If I have sex (behaviour) then it advantages my genes whether I believe I am having sex, cleaning the windows, praying, drinking beer or whatever. Now, unless we can show direct causation of behaviour by belief (such that I can only have sex if I believe I'm having sex) we are stuck without a selection mechanism for belief. As there are very, very many false beliefs that are possible for any situation and only the one true one the chance of us having a true belief is small. Thus naturalism and evolution should lead us to believe that our cognitive processes are not reliably formed. As this covers all beliefs we form on the basis of those cognitive processes it covers evolutionary naturalism. Evolutionary naturalism confounds the reliability of its own formulation’. Evolutionary naturalism cannot have been reliably formed, it has a "defeater".

Now, Plantinga's “defeater” is “undercutting”, it does not suggest that either N or E is false. Nor does it suggest that “N and E” is inconsistent. The defeater acts on the reasons for accepting N and E, Rn and Re. Not that the defeater suggests that these are false, rather that there is no good reason to suppose that are true.

Are these good reasons necessary? Allowing good reasons as necessary for Plantinga's conception of knowledge together with Plantinga's anti-naturalist argument leads to the conclusion that neither Rn nor Re can be known and, by extension neither can “N and E”. But Plantinga's argument is not that “N and E” cannot be known (as an absolute sceptic I would have no issue with that conclusion) but that “N and E” is “unreasonable”. If good reasons are necessary for reasonable belief then, for S to be reasonable in disbelieving in either Rn or Re there would have to be good reasons for believing in their negations. Any arguments for the truth of their negations would be arguments for their falsity, which Plantinga has not given. (I am ignoring issues of agnosticism. This is partly because I have powerful intuitions that widespread enforced agnosticism would cause problems. But it is mostly because I haven't figured out how to handle the issues of agnosticism).

Thus Plantinga fails to support his case. But worse, Plantinga's insistence that S rejects “N and E” commits her to an irrational belief. Given that S reasonably believes Rn and Re she is not only reasonable in believing R and E she would be unreasonable if she accepted Plantingas call to reject R and E.

S believes:

1.Rn2.Re

Plantinga offers no arguments against the entailment of naturalism and evolution by the reasons S has and they do, in fact, entail. Thus S also believes:

3.If Rn then N4.If Re then E

From 1. and 3., by modus ponens, S is committed to believing:

5.N

and, from 2. and 4., by modus ponens, S is committed to believing:

6.E

Finally, from 5. and 6. by conjunction, S is committed to believing:

7.N and EIf S believes either not-N or not-E then she believes a contradiction.

Wednesday, 25 February 2009

In Warrant: the Current Debate1, Alvin Plantinga criticises Bayesian Coherentism. Bayesian Coherentism is a new fangled way of dealing with an old idea: that we should be consistent in the probabilities we apply to propositions and should act on those propositions in accordance with their probability.

As Plantinga points out Bayesian Coherentism does at least acknowledge that we have degrees of belief in propositions. Plantinga, for example, believes “that Banff is in Scotland, that there was such a thing as the American Civil War, that I am more than ten years old, and that 7 + 5 = 12; and I believe these in ascending order of firmness.”2 Whilst many epistemologies lack the means to deal with these distinctions, Bayesianism simply attaches probability statements to them and introduces a precise method coping with diverse probabilities: Bayes Theorem (for an explanantion of the theorem even I can understand, see Eliezer Yudkowsky's)

All this is to the good. To the bad are the difficulties in applying probability to everyday reasoning and the applicability of Bayesian arguments for coherence, usually in the form of gambling theory, to people who may not want to bet. Almost as a coupe de grace Plantinga pulls out the Paradox of the Preface to argue that, despite the Bayesian's protestations, it is rational to believe propositions with contradictory probabilities.

Plantinga goes further than is necessary, he need not argue that we can believe propositions with probabilities that are incoherent because...well...applying probabilities to propositions is bollocks.

Propositional and Empirical Probability

I can say that the probability of the next throw of a fair die coming up six is 1/6. Whilst it can be argued that this expresses a degree of belief that the next throw of a fair die will come up six, it doesn't express the degree of belief I hold in the proposition “the next throw of a fair die coming up six is 1/6”. To make it a little clearer and to remove the need for inverted commas, let's call “the next throw of a fair die coming up six is 1/6” by a name: “Ichabod”. Let's name a non-probabilistic statement, say “there is a pint of beer in front of me”, “Ahitub”. I am pretty confident that there is a pint of beer in front of me, but not entirely sure. There is a degree of belief I have in Ahitub, despite the proposition itself containing no probability statement. Using the language of probability it can be said that I believe Ahitub to X%. Applying the language of probability to Ichabod, it can be said that I believe Ichabod to Y%. The probabilistic elements of proposition, the 1/6 probability predicted of the die, are the “empirical probability”. The probabilistic terminology applied to the degree of belief in the propositions themselves is the “propositional probability”. Now, how big is Y%?

It appears that I can vary a propositional probability by varying the probabilistic element within a treatment. Whilst I only believe “there is a pint of beer in front of me” is probable, I believe “there is probably a pint of beer in front of me” completely. Actual numbers, given a few auxiliary assumptions, can be given in the case of dice. Let us take it that a die is a cube, it has six equal sides. A “fair” die is so constructed that no one side is favoured over another when thrown and that its edges are so constructed that, when thrown, it must show one side and one side only.

If we cast the propositions in terms of empirical certainty there are six possible propositions:

1.The die, on the next throw, will come down “one”2.The die, on the next throw, will come down “two”3....etc

one of which must be true. Thus the combined probability of one of the six propositions being true is 1. Given that the die is a “fair” die the distribution of that probability over the propositions will be even: each empirically certain proposition is believed 1/6.

Given our auxiliary assumptions, the empirical probabilities of the six possible outcomes are:

1.The probability of the next throw coming up “one” is 1/62.The probability of the next throw coming up “two” is 1/63....etc.

the combination of which must come to 1. Something will happen and, as our auxiliary assumptions limit that something to one of the six I must believe “the summation of the probabilities is 1” with certainty. Our assumptions also determine that no individual outcome is preferred: I must also believe the distribution of that probability with certainty.

We either believe the empirically certain proposition “the die, on the next throw, will come down “six”” with a propositional probability of 1/6 or believe the empirical probability statement “the probability of the next throw coming up “six” is 1/6” with propositional certainty. I leave it for someone who has the mathematical ability to show that this holds for all values of empirical and propositional probability but, at the limits, it is clear that:

Claim 2: my propositional belief in Ichabod, described in terms of probability, is 100%

Unavoidable Propositional Uncertainty

So far, so good. However, let us consider another die. This die is supplied by 'Doc' Plantinga, a philosopher of international renown who has built up a lucrative sideline in scamming games of chance. This die is also a cube, it has six equal sides and is so constructed that when thrown, it must show one side and one side only. This die, however, is anything but a “fair” die. 'Doc' Plantinga has almost certainly loaded the die so that one side is favoured over the others when thrown. In predicting the likelihood of a particular number being thrown I must factor in the likelihood of that side being weighted for and the extent of the distortion (whether it biases the die to that number or guarantees that number).

Unfortunately, although I am pretty confident that 'Doc' has fixed the die I do not know in which direction. The same factor must be applied to the same original probability for each side and the result for all six sides must still add to 1. Thus the 'new' empirical probability is the same as the old: 1/6. From the two claims above it follows that my “propositional probability” for “the probability of 'Doc' Plantinga's die coming up six on the next throw” (call this proposition “Solomon”) is 100%.

Claim 3: Whatever 'degree of belief' I attach to a statement of empirical probability, the propositional probability of that statement is 100%

Whoever thought otherwise?

On reflection it is quite ridiculous to apply probabilities to our attitudes to propositions. Whilst we may estimate probabilities, our actions (both practical and intellectual) are binary: we either act as if we believe or we do not. We do not “probably” drink a pint of beer, nor do we press down really hard on the pencil if we particularly believe the proposition we are writing. All logics reduce to two value logic: that 1/6 chance either is, or isn't, 1/6 and it is either true or false to say of a variable in n-value logic that it stands at value n. Even in games of chance, deliberate exercises in allowing probabilistic judgements, we are rarely in a position to align our actions with our estimates of probabilities. We either bet on 'Dog's Dinner' in the 3:30 or do not, I am given one hand in poker and must either call, raise or fold.

So what is going on?

Our resort to probabilities results from their success elsewhere, the necessity of something to describe degrees of belief and the idea that probability is the only game in town. I do not believe Solomon to anywhere near the degree I believe Ichabod. What other concept can be used to distinguish my attitude to Ichabod from my attitude to the Solomon? What is different? It is my attitude to revising my beliefs in the light of future information. If the fair die came up six on each of the first three throws I would put this down to an unusual, but not outrageously unlikely, event and continue to believe that the odds of the next throw coming up six to be 1/6. If 'Doc' Plantinga's die came up six on each of the first three throws I would start to question whether this was the side he had weighted for and start to revise my estimate of the odds of the next throw with this die coming up six. Right now, if I believe Solomon, I believe it with the same probability as Ichabod but I believe it a darn sight less tenaciously.

This solves the “Paradox of the Preface”

As the Paradox of the Preface, which Plantinga uses to justify inconsistency, arises from the probability calculus removing propositional probability can be expected to solve the paradox. And so it does. I shall use Plantinga's formulation of the “Paradox of the Preface” as he casts it neatly in terms of belief, nicely isolating that part of the traditional view of knowledge:

“I write a book named I Believe! reporting therein only what I now fully believe. Being decently modest, I confess in the preface that I also believe that at least one proposition in I Believe! is false (although I have no idea which ones[s]). Then my beliefs are inconsistent, in the sense that there is no possible world in which they are all true; but might they not nevertheless be perfectly rational?”3

He is being perfectly rational and not at all inconsistent. Let us expand the statement “I believe X” to encompass two statements:

1.I have adopted X, for the present, and will act as if it is true2.I am happy to revise that belief in the light of future experience

And let us imagine a strangely sceptical Plantinga who believes just five propositions, P1, P2, P3, P4 and P5. The text I Believe! can be summarised:

1.I have adopted P1 and am prepared to revise it in the light of future experience2.I have adopted P2 and am prepared to revise it in the light of future experience3.I have adopted P3 and am prepared to revise it in the light of future experience4.I have adopted P4 and am prepared to revise it in the light of future experience5.(The Preface) I have adopted P1, P2, P3 and P4 and am prepared to revise P1, P2, P3 and P4 it in the light of future experience